
and (REF ) is the probability distribution of eigenvalues of GOE, GUE and GSE with \(\beta =1,2\)  and 4, respectively [1]}, [2]}. Such a Gibbs measure is called a Coulomb gas because of the logarithmic interactions.
Dyson's results of matrix-valued processes are generalized by Katori and Tanemura [3]}, and it is proved that when matrix entries are replaced with independent Brownian motions, their eigenvalue processes satisfy the SDEs (up to a time change):
\(d\lambda _i(t)=\sqrt{2}dB_i(t)+{\beta }\sum _{j(\ne i)}\frac{1}{\lambda _i(t)-\lambda _j(t)}dt,\ \ \beta >0,\ \ i=1,\dots ,N.\) 
